10,878 research outputs found
A Generalized Diagonal Wythoff Nim
In this paper we study a family of 2-pile Take Away games, that we denote by
Generalized Diagonal Wythoff Nim (GDWN). The story begins with 2-pile Nim whose
sets of options and -positions are and
\{(t,t)\mid t\in \M \} respectively. If we to 2-pile Nim adjoin the
main-\emph{diagonal} as options, the new game is
Wythoff Nim. It is well-known that the -positions of this game lie on two
'beams' originating at the origin with slopes
and . Hence one may think of this as if, in the process of
going from Nim to Wythoff Nim, the set of -positions has \emph{split} and
landed some distance off the main diagonal. This geometrical observation has
motivated us to ask the following intuitive question. Does this splitting of
the set of -positions continue in some meaningful way if we, to the game of
Wythoff Nim, adjoin some new \emph{generalized diagonal} move, that is a move
of the form , where are fixed positive integers and ? Does the answer perhaps depend on the specific values of and ? We
state three conjectures of which the weakest form is: exists, and equals , if and only if is a
certain \emph{non-splitting pair}, and where represents the
set of -positions of the new game. Then we prove this conjecture for the
special case (a \emph{splitting pair}). We prove the other
direction whenever . In the Appendix, a variety of experimental
data is included, aiming to point out some directions for future work on GDWN
games.Comment: 38 pages, 34 figure
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
Performances of conformal and planar arrays
Static and dynamic deformations can have a severe impact on the performance of conformal antennas on aircrafts and other vehicles. Therefore it is essential to study the different deformation and vibration mechanisms and their influence on the antenna's radiation pattern. This presentation gives an overview of different approaches concerning electromagnetic modelling of array antennas and investigations on antenna deformations presented in the scope of TG20
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